Lasers And Electro-Optics Research at the Cutting Edge 1600211941, 9781600211942, 9781606925294

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Steven B. Larkin Editor

Lasers and Electro-Optics Research at the Cutting Edge NOVA

LASERS AND ELECTRO-OPTICS RESEARCH AT THE CUTTING EDGE

LASERS AND ELECTRO-OPTICS RESEARCH AT THE CUTTING EDGE

STEVEN B. LARKIN EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2007 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data

ISBN 978-1-60692-529-4

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Chapter 1

Resonant Enhancement and Near-ELD Localizations of FS Pulses by Subwavelength NM-Size Metal Slits S.V. Kukhlevsky

Chapter 2

Single Experimental Setup for High Sensitive Absorption Coefficient and Optical Nonlinearities Measurements Debabrata Goswami

43

Chapter 3

Nonlinear Optics as a Method for Materials Chris J. Lee, Clare J. Strachan, Thomas Rades, Peter J. Manson

63

Chapter 4

Environmental Monitoring by Laser Radar Fiorani Luca

119

Chapter 5

Thermal Effects and Power Scaling of Diode-Pumped Solid-State Lasers Xiaoyuan Peng and Anand K. Asundi

173

Chapter 6

Catastrophe Optics in the Study of Spreading of Sessile Drops Nengli Zhang, David F. Chao and John M. Sankovic

193

Chapter 7

Recent Advances in TEA CO2 Laser Technology D.J. Biswas, J.P. Nilaya, M.B. Sai Prasad, and A. Kumar

213

Chapter 8

Novel Bismuth-Activated Glasses with Infared Luminescence Mingying Peng and Jianrong Qiu

253

Index

1

273

PREFACE It is expected that ongoing advances in optics will revolutionize the 21st century as they began doing in the last quarter of the 20th. Such fields as communications, materials science, computing and medicine are leaping forward based on developments in optics. This new book presents leading edge research on optics and lasers from researchers spanning the globe. The existence of resonant enhancement and near-field collimation of light waves by subwavelength apertures in metal films [for example, see T.W. Ebbesen et al., Nature (London) 391, 667 (1998) and H.J. Lezec et al., Science, 297, 820 (2002)] leads to the basic question: Can a light wave be enhanced and simultaneously localized in space and time by a subwavelength slit? To address this question, the spatial distribution of the electromagnetic field of an ultrashort (femtosecond) wave-packet scattered by a subwavelength (nanometersize) slit was analyzed by using the conventional approach based on the Neerhoff and Mur solution of Maxwell’s equations. The results show that a light wave can be resonantly enhanced by orders of magnitude and simultaneously localized in the near-field diffraction zone at the nm and fs scales. Chapter 1 includes many illustrations to facilitate an understanding of the natural spatial and temporal broadening of light beams and the physical mechanisms that are contributing to the resonantly enhanced scattering and localization of fs pulses by subwavelength nm-size metal slits. The results are discussed in the context of possible applications in the near-field scanning optical microscopy (NSOM). As explained in chapter 2, accurate knowledge of absorption coefficient of a sample is a prerequisite for measuring the third order optical nonlinearity of materials, which can be a serious limitation for unknown samples. The authors introduce a method, which measures both the absorption coefficient and the third order optical nonlinearity of materials with high sensitivity in a single experimental arrangement. They use a dual-beam pump-probe experiment and conventional single-beam z-scan under different conditions to achieve this goal. They also demonstrate a counterintuitive coupling of the non-interacting probe-beam with the pump-beam in pump-probe z-scan experiment. With the development of stable, compact and reliable pulsed laser sources the field of characterising materials, and in particular interfaces through their nonlinear optical response has bloomed. Second harmonic generation due to symmetry breaking at interfaces has provided a new spectroscopic tool of great utility. For example, it has been used to observe diffusion processes in semiconductors and to monitor the surface concentration of glucose oxidase in water. The surface response of various stained cellular features has been used as a source of illumination in microscopy, providing results similar to fluorescence microscopy but with less absorption and resulting damage.

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The nonlinear optical response of various materials provides a very sensitive technique for measuring and monitoring some phase changes such as crystallisation from solution, micellisation of surfactants and polymorphic transitions of explosives. The work has focused on characterising and determining concentrations of pharmaceutically interesting bulk compounds, dispersions and emulsions. This work has potential application for in-line monitoring and quality control of pharmaceutical manufacturing. In chapter 3 the authors present an extensive review of various spectroscopic techniques that make use of the nonlinear optical response of one or more media. They also present the results of our own work in this field. Since the discovery of laser, optical radars or lidars have been successfully applied to the monitoring of the three main environments of our planet: lithosphere, hydrosphere and atmosphere. The transmission of light depends on the medium: while in soil it does not propagate, in water and air it can typically travel for meters and kilometers, respectively. Analogously, three theoretical frameworks can be established: lidar equations for hard, dense and transparent targets. Such different behaviors drive the range of lidar applications in the three above mentioned environments. In the lithosphere, laser radars have been applied to three-dimensional scans of underground cavities. In the hydrosphere, lidar fluorosensors have been very effective in the bio-optical characterization of the first layers of sea waters. Such instruments are usually aboard planes and ships and can help filling the gap between in situ measurements and satellite imagery. The medium where lidars are unbeaten, at least for some purposes, is air. Atmospheric applications of laser radars range from troposphere (e.g. pollution monitoring in urban areas and wind speed measurements) to stratosphere (e.g. polar stratospheric cloud detection and ozone hole assessment), and even mesosphere (e.g. profiling of K and Na). The purpose of Chapter 4 is twofold: from one hand, the interested reader is introduced in lidar science and technology, to the other one, the researcher familiar with laser remote sensing is faced with some current investigations. The first aim is achieved by introducing the lidar principles, for hard, dense and transparent targets, and by illustrating selected case studies, taken from the experience of the author, in order to exemplify some relevant applications of such principles. The second one is pursued by describing in more detail the most recent results obtained by the author in the field of environmental monitoring by laser radar. In Chapter 5, thermal effects of diode-pumped solid-state (DPSS) lasers were theoretically analyzed and modeled for Nd doped lasing materials. The models determined the thermal lens and separated the end effect from the thermal effects. The knowledge of thermal lensing in the laser cavity is critical to scale the laser output power with the diffraction-limit laser beam output and to optimize the design of DPSS lasers. To validate the model, a new modified Twyman-Green interferometer is proposed to measure the thermal effects directly and accurately. Good agreement with the analytical model was obtained. Based on these pre-determined thermal lens effects, a power-scaling model was setup to guide the development of DPSS lasers. Within the fracture limits of the lasing crystals under highpower end-pumping, together with the design aims of the TEM00 mode output and the conversion efficiency, the model optimized the mode-to-pump ratio, doping concentration and dimensions of the lasing crystals. It also predicted the output power of DPSS lasers. The model showed a good agreement with the experiment later on. Based on the model, a practical diode-pumped Nd:YVO4 laser system with a simple linear cavity was demonstrated. The laser produced an output power of 9.8W in the TEM00 mode under a pump power of 28W

Preface

ix

where the rod surface acting as one of end mirrors in the linear plano-concave resonator. In conclusion, the models are quite useful for broadly guiding the design and development of most DPSS lasers, including rare earth ions doped YAG, YVO4, and YLF, etc. As presented in Chapter 6, optical catastrophe occurring in far field of waves of droprefracted laser beam implies a wealth of information about drop spreading. When a parallel laser beam passes through a sessile drop on a slide glass, the rays are refracted by the drop and produce a new wave field in space after the drop. Very interesting optical image patterns occur on a screen far from the drop. Stable spreading-and-evaporations of sessile drops on an isothermal and isotropic slide-glass surface give nearly perfect circular far-field sectional images of the drop-refracted laser beam on the distant screen with a bright-thick ring and a set of fringes. The image configurations are time dependent and sensitive to the drop properties. In some cases of evaporating drops, rapidly varying optical image patterns occur on the screen. Classical geometric optics cannot correctly interpret these optical image patterns. An analysis based on catastrophe optics reveals and interprets the formation of these optical image patterns. The circular caustic line manifested as the bright-thick ring on the screen implies that the caustic is the lowest hierarchy of optical catastrophes, called fold caustic, which is produced from the initial wavefront with a local third-order surface given by the drop refraction. The set of fringes next to the fold caustic is the caustic diffraction instead of an aperture effect misunderstood by many investigators. The caustics and their diffraction fringes on the far-field sectional images of the drop-refracted laser beam can sensitively detect the interface instability of sessile drops, measure real-time local contact angles, estimate the drop-foot heights, visualize thermo-capillary convection flows, and investigate the spreading characteristics of evaporating liquid drops, such as drop profiles, effects of evaporation and thermo-capillary convection flows on the spreading, etc. The successful applications of catastrophe optics to the study of drop spreading convincingly show that catastrophe optics is not only able to qualitatively explain optical phenomena in nature but also able to quantitatively measure physical parameters of the objects of study, such as spreading drops. The new results obtained in the area of repetitive and single longitudinal mode operation of TEA CO2 lasers have been presented in Chapter 7. Helium-free operation of a conventional TEA CO2 laser excited by a novel pulser circuit and its advantages in repetitive and single longitudinal mode lasing has been described. Much higher detunability of the single longitudinal mode achieved by making use of a traveling wave cavity in conjunction with a saturable absorber is discussed. A latch-proof pulser driven by an indigenously developed rotating dielectric spark gap switch makes the repetitive operation possible. Chapter 8 discusses broadband infrared luminescence covering the optical telecommunication wavelength region of O, E, S, C and L bands (1200~1600nm), with FWHM (Full Width at Half Maximum) of 200~400nm and fluorescent lifetime of 200~700μs, which was observed from bismuth-doped SiO2-Al2O3, GeO2-Al2O3, GeO2-B2O3, GeO2-Ga2O3, GeO2-Ta2O5, SiO2-B2O3, SiO2-Ta2O5, Al2O3-B2O3, Al2O3-P2O5 and ZnOAl2O3-SiO2 glass systems at room temperature in the case of 808nm excitation. In the absorption spectra of these glasses, a strong absorption peak near ~500nm was always detected in all samples, while absorption peaks at ~700, ~800 and ~1000nm were only in some of hosts such as GeO2-M2Ox glasses where M stood for B, Al, Ga and Ta. As usual, the emission and excitation peaks for Bi3+-doped luminescent materials are located in ultraviolet or visible wavelength region, and the radiative lifetime seldom exceeds 10μs. Thus, the

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Steven B. Larkin

luminescent properties of these Bi-doped glasses are distinguishably different from the Bi3+ doped luminescent materials previously reported in literatures. Compared with the experimental results on BiO molecule in gas phase reported by Fink et al and Shestakov et al, we tentatively assign the broadband infrared luminescence to the BiO molecule dissolved in these bismuth doped vitreous matrices. The observed absorptions at ~500nm, 700nm, 800nm and 1000nm could then be attributed to the transitions of X1→(H, I), X1→A2, X1→A1 and X1→X2 respectively, and the emission peak at 1300nm could be to X2→X1. These glasses may have potential applications in widely tunable laser and super-broadband optical amplifier for the optical communications.

In: Lasers and Electro-Optics Research at the Cutting Edge ISBN 1-60021-194-1 c 2007 Nova Science Publishers, Inc. Editor: Steven B. Larkin, pp. 1-42

Chapter 1

R ESONANT E NHANCEMENT AND N EAR -F IELD L OCALIZATION OF F S P ULSES BY S UBWAVELENGTH N M -S IZE M ETAL S LITS S. V. Kukhlevsky 1Institute of Physics, University of Pcs, Ifjsg u. 6, Pcs 7624, Hungary

Abstract The existence of resonant enhancement and near-field collimation of light waves by subwavelength apertures in metal films [for example, see T.W. Ebbesen et al., Nature (London) 391, 667 (1998) and H.J. Lezec et al., Science, 297, 820 (2002)] leads to the basic question: Can a light wave be enhanced and simultaneously localized in space and time by a subwavelength slit? To address this question, the spatial distribution of the electromagnetic field of an ultrashort (femtosecond) wave-packet scattered by a subwavelength (nanometer-size) slit was analyzed by using the conventional approach based on the Neerhoff and Mur solution of Maxwell’s equations. The results show that a light wave can be resonantly enhanced by orders of magnitude and simultaneously localized in the near-field diffraction zone at the nm and fs scales. The chapter includes many illustrations to facilitate an understanding of the natural spatial and temporal broadening of light beams and the physical mechanisms that are contributing to the resonantly enhanced scattering and localization of fs pulses by subwavelength nm-size metal slits. The results are discussed in the context of possible applications in the near-field scanning optical microscopy (NSOM).

1.

Introduction

In the last decade nanostructured optical elements based on scattering of light waves by subwavelength-size metal objects, such as particles and screen holes, have been investigated, intensively. The most impressive features of the optical elements are resonant enhancement and spatial localization of optical fields by the excitation of electron waves in the metal (for example, see the studies [1–76]). In the last few years, a great number of studies have been devoted to the nanostructures in metal films, namely a single aperture, a grating of apertures and an aperture surrounded by grooves. The study of collimation and

2

S.V. Kukhlevsky

resonant enhanced transmission of waves in close proximity to a single subwavelength aperture acting as a microscope probe [1–5] was connected with developing near-field scanning microwave and optical microscopes with subwavelength resolution [6–9]. The resonant transmission of light by a grating with subwavelength apertures or a subwavelength aperture surrounded by grooves is an important effect for nanophotonics [10–76]. The transmission, on the resonance, can be orders of magnitude greater than out of the resonance. It was understood that the enhancement effect has a two-fold origin: First, the field increases due to a pure geometrical reason, the coupling of incident plane waves with waveguide mode resonances located in the slit, and further enhancement arises due to excitation of coupled surface plasmon polaritons localized on both surfaces of the slit (grating) [16, 38, 40]. A dominant mechanism responsible for the extraordinary transmission is the resonant excitation of the waveguide mode in the slit giving a Fabry-Perot like behavior [38]. In addition to the extraordinary transmission, a series of parallel grooves surrounding a nanometer-size slit can produce a micrometer-size beam that spreads to an angle of only few degrees [15]. The light collimation, in this case, is achieved by the excitation of coupled surface plasmon polaritons in the grooves [40]. At appropriate conditions, a single subwavelength slit flanked by a finite array of grooves can act as a ”lens” focusing a light wave [54]. It should be noted, in this connection, that the diffractive spreading of a beam can be reduced also by using a structured aperture or an effective nanolens formed by self-similar linear chain of metal nanospheres [55, 56]. New aspects of the problem of resonantly enhanced transmission and collimation of light are revealed when the nanostructures are illuminated by an ultra-short (fs) light pulse [57–62]. For instance, in the study [57], the unique possibility of concentrating the energy of an ultrafast excitation of an ”engineered” or a random nanosystems in a small part of the whole system by means of phase modulation of the exciting fs-pulse was predicted. The study [58] theoretically demonstrated the feasibility of nm-scale localization and distortionfree transmission of fs visible pulses by a single metal slit, and further suggested the feasibility of simultaneous super resolution in space and time of the near-field scanning optical microscopy (NSOM). The quasi-diffraction-free optics based on transmission of pulses by a subwavelength nano-slit has been suggested to extend the operation principle of a 2-D NSOM to the ”not-too-distant” field regime (up to ∼ 0.5 wavelength) [59]. Some interesting effects namely the pulse delay and long leaving resonant excitations of electromagnetic fields in the resonant-transition gratings were recently described in the studies [41, 60]. The great interest to resonant enhanced transmission, spatial localization (collimation) of continuous waves and light pulses by subwavelength metal apertures leads to the basic question: Can a light pulse be enhanced and simultaneously localized in space and time by a subwavelength slit? If the field enhancement can be achieved together with nm-scale spatial and fs-scale temporal localizations, this could greatly increase a potential of the nanoslit systems in high-resolution applications, especially in near-field scanning microscopy and spectroscopy. In the present study we test whether the resonant enhancement could only be obtained at the expense of the spatial and temporal broadening of a light wavepacket. To address this question, the spatial distribution of the energy flux of an ultrashort (fs) pulse diffracted by a subwavelength (nano-size) slit in a thick metal film of perfect conductivity will be analyzed by using the conventional approach based on the Neerhoff and Mur solution of Maxwell’s equations. The chapter is organized as follows. In short, we first

Resonant Enhancement and Near-Field Localization of FS Pulses...

3

will discuss the natural spatial and temporal broadening of light beams in free-space (Section 2.). The theoretical development of Neerhoff and Mur will be described in Section 3. and the model will then be used to calculate the spatial distribution of the electromagnetic field of the continuous wave (Section 4.) and pulse (Section 5.) scattered by the subwavelength slit under various regimes of the near-field diffraction. We will show that a light can be enhanced by orders of magnitude and simultaneously localized in the near-field diffraction zone at the nm and fs scales. The results presented in Sections 4. and 5. are discussed in the context of possible applications in the near-field scanning optical microscopy (NSOM). In Section 6., we summarize results and present conclusions.

2.

Natural Spatial and Temporal Broadening of Light Beams

In Section 2., we consider several general properties of the optical beams, which are important for understanding the central topic of this chapter, the resonant enhancement and near-field localization of fs pulses by subwavelength nm-size metal slits. The material of the section is reprinted from Optics Communications, Vol. 209, S.V. Kukhlevsky and G. Nyitray, Correlation between spatial and temporal uncertainties of a wave-packet, 377-382, Copyright (2002), with permission from Elsevier. The section discuses the physical mechanisms that are contributing to the natural spatial and temporal broadening of continuous waves and light pulses in free space. The natural spatial and temporal broadening of light beams in free space are closely connected with the spatial and temporal uncertainties of a wave-packet. The spatial (∆x ∼ 1/∆k) and temporal (∆t ∼ 1/∆ω) uncertainties of a wave-packet are the universal laws of physics, and ones of the best understood. The detailed description of the phenomena can be found in optical textbooks. The uncertainty relations affect all classical and quantum-mechanical fields without exception. For instance, given the de Broglie postulate p=¯hk, the spatial uncertainty in quantum mechanics is determined by the Heisenberg relation ∆x ∼ ¯ h/∆p. Because of the uncertainty it is impossible to define at a given time both the position ∆x of a wave packet and its direction ∆p/p to an arbitrary degree of accuracy. The temporal uncertainty (∆t ∼ 1/∆ω), which in quantum mechanics is often called the fourth Heisenberg uncertainty relation ( ∆t ∼ ¯ h/∆E), shows that for a wave packet, the smaller the frequency uncertainty ∆ω, the more long the packet duration ∆t. The mathematical principle behind the spatial and temporal uncertainties of wave packets involves a well-known relation between the widths of two functions, which are Fourier transforms of each other. Owing to the relation, the greater the energy uncertainty ∆E of a quantum system, the more rapid the time evolution ∆t. Another example is diffraction phenomenon: the limitation of the phase space (∆k) of a wave by the xap. and yap. boundaries of an aperture causes the spatial broadening of the wave in the {x}{y} domain. According to the space-frequency uncertainty, the beam angle divergence ∆k/k increases with decreasing the region of spatial localization ∆x. Similarly, the decrease of the spectral region ∆k causes the increase of the value ∆x. According to the quantum-mechanics operator formalism, the spatial and temporal uncertainties arise because the conjugate variables must satisfy the Heisenberg postulate. The spatial ∆x and temporal ∆t uncertainties of a wave packet are usually considered to be independent each other. In the frames of such consideration the unlimited small

4

S.V. Kukhlevsky II

x λ = (2/3)∆z

I

III

λ = ∆z λ = 2∆z

Incident w.-packet

-∆z 2

+∆z 2

Output w.-packet z

Bounded w.-packet

Boundary

Boundary

Figure 1. Propagation of a wave-packet through the bounded space. temporal and spatial uncertainties (∆t → 0, ∆x → 0) of a wave packet can be achieved simultaneously. This can be realized, for instance, by using an ultra-short wave-packet (∆t → 0, ∆ω → ∞) passing through a small aperture (∆x → 0, ∆k → ∞). The value of the central wavelength λ0 of the wave packet can be arbitrary. The following simple consideration shows that the spatial (∆x ∼ 1/∆k) and temporal (∆t ∼ 1/∆ω) uncertainties of a wave-packet correlate with each other, the smaller uncertainty ∆x, the bigger uncertainty ∆t. The value of the uncertainty ∆t is determined by the width ∆x and the value of the central wavelength λ0 of the wave packet. Let us investigate propagation of a wave packet of light through the bounded space. For the sake of simplicity, we consider an one-dimensional scalar packet passing through the space confined by two boundaries, as shown in Fig. 1. At a point P (x, z) of the region I, an incident wave-packet ψ(P, t) of the duration ∆tI can be presented in the form of the Fourier time expansion: ψ(P, t) =

Z ∞

ψ(P, ω) exp(−iωt)dω,

(1)

0

where ω = ckz is the light frequency. The propagation of the Fourier components ψ(P, ω(kz )) in the regions I, II and III is governed by the Helmholtz equation [∇2z + kz2 ]ψ(P, kz ) = 0,

(2)

with boundary conditions imposed. In the non-bounded regions I and III, the Fourier spectra is continuous with the value ω(kz ) ∈ [0, ∞]. In the confined space II, the values of momenta kz = πn/∆z are discrete (n = 1, 2, ...) and a nonzero lower limit kzmin = π/∆z exists due to the boundaries (see, Fig. 1). Compared with the regions I and III, the spectral width in the bounded region II is smaller, ω ∈ [ω(kzmin ), ∞]. According to the Fourier analysis, the decrease of the spectral width ∆ω of the wave packet in the region II (the spectral interval changes from [0, ∞] to [ω(kzmin ), ∞] causes increasing of the packet duration ∆tII,III in

Resonant Enhancement and Near-Field Localization of FS Pulses...

5

1,0

Amplitude

0,5

A

B

0,0

-0,5

-1,0 -10

-5

0

5

10

Ti me ( f s )

(a)

ω0 ωco No r ma l i z e d Amp l i t u d e

1,0 0,8 0,6 0,4

B

A

0,2 0,0 17,5

20,0

22,5

25,0

27,5

30,0

15

F r e q u e n c y ( 10 Hz )

(b) Figure 2. (a) The amplitudes of the original (A) and modified (B) wave-packets. (b) The Fourier spectra of the original (A) and modified (B) wave-packets. Here, ψ(P, t) = exp[−2ln(2)(t/τ )2]exp(iω0t); τ = 1 fs is the wave-packet duration; ω0 is the central frequency of the wave packet. The figure demonstrates the broadening of the fs pulse by cutting off the wave-packet spectra at the frequency ωco .

6

S.V. Kukhlevsky

the regions II and III. As an example, Fig. 2. demonstrates the increasing of the width of a femtosecond wave-packet by cutting off the wave-packet spectra at the frequency ωco = ω(kzmin ). The cut-off frequency ωco depends on the boundary conditions. The approximate value is given by ω(kzmin ) = cπ/∆z. The cut-off frequency increases with decreasing of the width ∆z of the packet. Thus, the temporal uncertainty ∆tII,III is limited by the spatial uncertainty ∆z. Such behavior indicates the correlation between the spatial ∆z ∼ 1/∆kz and the temporal ∆t ∼ 1/∆ω uncertainties, the smaller uncertainty ∆z, the bigger uncertainty ∆t. In the case of a wave-packet confined in three directions, the nonzero lower limits of the momenta k at x, y and z directions are given respectively by kxmin = π/∆x, kymin = π/∆y and kzmin = π/∆z, and the cut-off frequency is given by ω(kmin ) = c((π/∆x)2 + (π/∆y)2 + (π/∆z)2)1/2.

(3)

It can be easily demonstrated, using Eq. 3 and the relation ∆t ∼ 1/∆ω, that the spacetime uncertainty of a wavepacket is given by ∆t ∼ 1/(ω0 + ∆ω/2 − c((π/∆x)2 + (π/∆y)2 + (π/∆z)2 )1/2),

(4)

where ∆t is the duration of the spatially localized wave-packet, and ω0 and ∆ω are the central frequency and the spectral width of the free-space (unbounded) packet. The temporal uncertainty ∆t of a wave-packet increases with decreasing the spatial uncertainties ∆x, ∆y and ∆z. The increase of the temporal uncertainty ∆t of the wave packet causing by decreasing its spatial uncertainty ∆x can be reduced by increasing the central frequency ω0 of the packet. In the limit of ω0 →∞, the light wave-packets are described by geometrical optics. We believe that the space-time uncertainty relation (4) can be considered as a general property of a wave-packet, because it is based on the basic properties of the Fourier transformation. The spatial, temporal and space-time uncertainty relations for a light wave are in agreement with the results of exact analytical and numerical solutions of Maxwell’s equations for both the under-wavelength (Λ > λ) and subwavelength (Λ < λ) domains. Here, Λ is the characteristic transversal dimension of the wave. For an example, in the case of a plane wave E(z, t)∼E(t) exp(ikz z), the pulse E(t) of the duration ∆t corresponds to the field E(ω) that is localized in the Fourier spectral region ∆ω. The regions of localization ∆t and ∆ω satisfy the temporal uncertainty relation ( ∆t ∼ 1/∆ω). In the under-wavelength domain, the angle divergence ∆k/k determined by the spatial uncertainty ∆x ∼ 1/∆k is in agreement with the diffractive broadening of a beam predicted by the Fresnel-Kirchhoff diffraction theory [63]. Because of the spatial uncertainty, light spreads out in all directions after passing a metal slit smaller than its wavelength as shown in Fig. 3 (see, for example Refs. [1–9]). The harder one tries to decrease the beam transverse dimension by narrowing the slit, the more it broadens out. Similarly, the beam width dramatically increases with increasing the distance from the slit. The spatial uncertainty imposes a fundamental limit on the transverse dimension of a beam at a given distance from a subwavelength aperture and consequently limits the resolution capabilities and makes harder the position requirements of subwavelength-beam optical devices, such as near-field scanning optical microscopes (NSOM) and spectroscopes (see, Refs. [1–9]). One of the most important consequences

Resonant Enhancement and Near-Field Localization of FS Pulses...

7

x Metal film

I

Diffracted wave

a b II

k

z E Incident wave

-a

Sample

H

ey 0 III Metal film

Figure 3. Diffraction of a continuous wave by a subwavelength nm-sized slit (waveguide) in a thick metal film and its implication to near-field optical microscopy. The slit width and length are 2a and b, respectively. of the space-time uncertainty is that the duration ∆t of a light pulse localized in the space increases with decreasing the region of spatial localization ∆x. In the under-wavelength domain, for an example, such a behavior is well known for the Gaussian beams [63, 64]. Although, the exact values of ∆t and ∆x of a light pulse scattered (localized) by a subwavelength aperture can be computed for the particular experimental conditions by solving the Maxwell equations, the correlation between the spatial and temporal uncertainties of the light wave-packet can be demonstrated without the computer simulation. Indeed, according to the subwavelength diffraction theory [1–5], the Fourier components ψ(P, ω) (continuous wave) having the frequency lower than the cut-off frequency ( ω < ωco (z, ∆x)) does not contribute energy to the components ψ ′(P ′, ω) of the diffracted wave-packet at a given distance z from the aperture. Thus, the effective duration ∆t(z = 0) of the wave-packet at the distance z = 0 increases due to the decrease of the effective spectral band width from [0, ∞] to [ωco , ∞]. The effective increase of the duration ∆t(z = 0) implies an increase of the pulse duration ∆t′ (z > 0) at a distance z, in other words, an increase in the temporal uncertainty associated with the incident wave-packet. The value of ωco increases with decreasing of the width ∆x and increasing of the distance z from the aperture [1–5]. Thus, the temporal uncertainty ∆t′ is limited by the spatial uncertainty ∆x′ = ∆x′ (z, ω, ∆x). The transverse dimension ∆x′ of the diffracted wave-packet increases with increasing of the distance z and the value 1/∆x. Such behavior indicates the correlation between the spatial ∆x ∼ 1/∆k and temporal ∆t ∼ 1/∆ω uncertainties of the light wave-packet, which could be observed in the subwavelength diffraction experiment. It can be mentioned that the ”cut-off frequency” phenomenon, in fact, is the Faraday screening of electromagnetic waves. The phenomenon is well-known also in theory of short waveguides, which can be considered as the limit of thick aperture. It should be noted that the time-space uncertainty correlation should be not limited to the case of diffraction on an aperture. Owing to Babinett’s principle, the similar behavior is expected for the diffraction on a small particle (Rayleigh-Mie scattering). Thus, the correlation should affect the fs-scale formation

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S.V. Kukhlevsky

of highly enhanced fields localized in nm-size regions of random clusters, composites and rough surfaces [23]. The phenomenon probably will be important also for the future experiments on the scattering of attosecond and zeptosecond light pulses by electrons, neutrons and protons [22]. The spatial and space-time uncertainty relations are in agreement with the results of an exact numerical model presented in Section 3., which demonstrate respectively the spatial (Section 4.) and spatio-temporal (Section 5.) broadenings of a continuous wave and a pulse scattered by a subwavelength nm-size metal slit.

3.

Model

An adequate description of the scattering of light through a subwavelength nano-sized slit in a metal film requires solution of Maxwell’s equations with complicated boundary conditions. The light-slit interaction problem even for a continuous wave can be solved only by extended two-dimensional (x, z) numerical computations. The three-dimensional (x, z, t) character of the pulse-slit interaction makes the numerical analysis even more difficult. Let us consider the near-field diffraction of an ultrashort pulse (wave-packet) by a subwavelength slit in a thick metal screen of perfect conductivity by using the conventional approach based on the Neerhoff and Mur solution of Maxwell’s equations. Before presenting a treatment of the problem for a wave-packet, we briefly describe the Neerhoff and Mur formulation [3, 5] for a continuous wave (a Fourier ω-component of a wavepacket). The transmission of a plane continuous wave through a slit (waveguide) of width 2a in a screen of thickness b is considered. The perfectly conducting nonmagnetic screen placed in vacuum is illuminated by a normally incident plane wave under TM polarization (magneticfield vector parallel to the slit), as shown in Fig. 3 The magnetic field of the wave is assumed to be time harmonic and constant in the y direction: ~ H(x, y, z, t) = U (x, z)exp(−iωt)~ey .

(5)

The electric field of the wave (5) is found from the scalar field U (x, z) using Maxwell’s equations: Ex (x, z, t) = −

ic ∂z U (x, z)exp(−iωt), ωǫ1

Ey (x, z, t) = 0.

Ez (x, z, t) =

ic ∂x U (x, z)exp(−iωt). ωǫ1

(6)

(7)

(8)

Notice that the restrictions in Eq. 5 reduce the diffraction problem to one involving a single scalar field U (x, z) in two dimensions. The field is represented by Uj (x, z) (j=1,2,3 in each of the three regions I, II and III). The field satisfies the Helmholtz equation: (∇2 + kj2 )Uj = 0,

(9)

Resonant Enhancement and Near-Field Localization of FS Pulses...

9

where j = 1, 2, 3. In region I, the field U1 (x, z) is decomposed into three components: U1 (x, z) = U i (x, z) + U r (x, z) + U d (x, z),

(10)

each of which satisfies the Helmholtz equation. U i represents the incident field, which is assumed to be a plane wave of unit amplitude: U i (x, z) = exp(−ik1z).

(11)

U r denotes the field that would be reflected if there were no slit in the screen and thus satisfies U r (x, z) = U i (x, 2b − z).

(12)

U d describes the diffracted field in region I due to the presence of the slit. With the above set of equations and standard boundary conditions for a perfectly conducting screen, a unique solution exists for the diffraction problem. To find the field, the 2-dimensional Green’s theorem is applied with one function given by U (x, z) and the other by a conventional Green’s function: (∇2 + kj2 )Gj = −δ(x − x′ , z − z ′ ),

(13)

where (x, z) refers to a field point of interest; x′ , z ′ are integration variables, j = 1, 2, 3. Since Uj satisfies the Helmholtz equation, Green’s theorem reduces to U (x, z) =

Z

Boundary

(G∂n U − U ∂n G)dS.

(14)

The unknown fields U d (x, z), U3(x, z) and U2 (x, z) are found using the reduced Green’s theorem and boundary conditions on G ǫ1 U (x, z) = − ǫ2 d

for b < z < ∞,

ǫ3 U3 (x, z) = ǫ2

Z a

−a

Z a

−a

G1(x, z; x′, b)DUb(x′ )dx′

G3 (x, z; x′, 0)DU0(x′)dx′

(15)

(16)

for −∞ < z < 0, U2 (x, z) = −

Z a

+

−a Z a

[G2 (x, z; x′, 0)DU0(x′) − U0 (x′)∂z′ G2 (x, z; x′, z ′)|z→0+ ]dx′

−a

[G2(x, z; x′, b)DUb(x′) − Ub (x′)∂z′ G2(x, z; x′, z ′)|z→b− ]dx′

(17)

for |x| < a and 0 < z < b. The boundary fields in Eqs. 15-17 are defined by U0 (x) ≡ U2 (x, z)|z→0+ ,

(18)

DU0 (x) ≡ ∂z U2 (x, z)|z→0+ ,

(19)

DUb (x) ≡ ∂z U2 (x, z)|z→b− .

(21)

Ub (x) ≡ U2 (x, z)|z→b− ,

(20)

10

S.V. Kukhlevsky In regions I and III the two Green’s functions in Eqs. 15 and 16 are given by i (1) (1) [H (k1R) + H0 (k1R′ )], 4 0 i (1) (1) G3(x, z; x′, z ′) = [H0 (k3R) + H0 (k3R′′)], 4 G1(x, z; x′, z ′) =

(22) (23)

with

′

R = [(x − x′ )2 + (z − z ′ )2 ]1/2, ′ 2

′

2 1/2

R = [(x − x ) + (z + z − 2b) ] ′′

′ 2

(24)

,

(25)

′ 2 1/2

(26)

R = [(x − x ) + (z + z ) ]

,

(1)

where H0 is the Hankel function. In region II, the Green’s function in Eq. 17 is given by i X∞ i exp(iγ0|z − z ′ |) + γ −1 4aγ0 2a m=1 m ×cos[mπ(x + a)/2a]cos[mπ(x′ + a)/2a]

G2(x, z; x′, z ′) =

×exp(iγm|z − z ′|),

(27)

where γm = [k22 −(mπ/2a)2]1/2. The field can be found at any point once the four unknown functions in Eqs. 18-21 have been determined. The functions are completely determined by a set of four integral equations: 2Ubi (x) −

ǫ1 Ub (x) = ǫ2

U0 (x) =

1 Ub (x) = − 2

Z a

−a

ǫ3 ǫ2

Z a

−a

Z a

−a

G1(x, b; x′, b)DUb(x′)dx′,

G3(x, 0; x′, 0)DU0(x′)dx′,

Z a

−a

Z a

−a

(29)

[G2(x, b; x′, 0)DU0(x′ ) − U0 (x′)∂z′ G2(x, b; x′, z ′)|z′ →0+ ]dx′ +

1 U0(x) = 2

(28)

[G2(x, b; x′, b)DUb(x′)]dx′,

(30)

[G2(x, 0; x′, b)DUb(x′) − Ub (x′ )∂z′ G2(x, 0; x′, z ′)|z′ →b− ]dx′ −

Z a

−a

[G2(x, 0; x′, 0)DU0(x′)]dx′,

(31)

where |x| < a, and Ubi (x) = exp(−ik1b).

(32)

The coupled integral equations 28-31 for the four boundary functions are solved nu~ ~ merically. The magnetic H(x, z, t) and electric E(x, z, t) fields of the diffracted wave in

Resonant Enhancement and Near-Field Localization of FS Pulses...

11

~ region I, II and III are found by using Eqs. 15-17. The magnetic H(x, z, t) fields in regions I, II, and III are given by H 1(x, z) = exp(−ik1 z) + exp(−ik1(2b − z))

N q ia ǫ1 X 1 H (k1 (x − xj )2 + (z − b)2)(DUb )j , − N ǫ2 j=1 0

H 2(x, z) = −

i 2N

q

k22

N √ 2 X (DU0)j + ei k2 |z| j=1

(33)

i q

2N k22

N N √ 2 X 1 i√k2 |z| X 1 (DUb )j − (U0)j + e 2 ×ei k2 |z−b| 2N 2N j=1 j=1

N ∞ √ 2 X 1 i X mπ(x + a) iγ1 |z| (Ub )j − cos e ×ei k2 |z−b| N γ 2a 1 m=1 j=1

×

N X

cos

j=1

×eiγ1 |z| × cos + ×

∞ mπ(x + a) 1 X mπ(xj + a) cos (DU0 )j − 2a N m=1 2a

N X

j=1

cos

∞ mπ(xj + a) 1 iγ1 |z−b| i X (U0)j + e 2a N m=1 γ1

N mπ(xj + a) mπ(x + a) X cos (DUb)j 2a 2a j=1

∞ x + a iγ1 |z−b| 1 X cos(mπ )e N m=1 2a N X

j=1

cos

mπ(xj + a) (Ub )j , 2a

H 3(x, z) = iǫ3

N X a

j=1 (1) ×H0



q

N ǫ2

(34)

~ 0)j (DU

k3 (x − xj

)2

+

z2



,

(35) (1)

where xj = 2a(j − 1/2)/N − a, j = 1, 2, . . ., N ; N > 2a/z; H0 (X) is the Hankel func~ i = H i · ~ey , i = 1, 2, 3. The coefficients (DU ~ 0 )j are found by solving numerically tion; H the four coupled integral equations (28-31). The electric field of the wave is found from the scalar field U (x, z) using Eqs. 6-8. For instance, the electrical field in the region III is given by Ex3(x, z, t) = −

N z a (ǫ3)1/2 X (1) H [k3((x − xj )2 + z 2 )1/2] N ǫ2 j=1 ((x − xj )2 + z 2 )1/2 1

~ 0)j exp(−iωt), ×(DU

(36)

12

S.V. Kukhlevsky Ey3(x, z, t) = 0,

Ez3(x, z, t) =

(37)

N a (ǫ3 )1/2 X x − xj (1) H [k3((x − xj )2 + z 2 )1/2] N ǫ2 j=1 ((x − xj )2 + z 2 )1/2 1

~ 0)j exp(−iωt). ×(DU

(38)

For more details on the model and the numerical solution of the Neerhoff and Mur coupled integral equations, see the references [3, 5]. Let us now consider the scattering of an ultra-short pulse (wave packet). The magnetic field of the incident pulse is assumed to be Gaussian-shaped in time and both polarized and constant in the y direction: ~ H(x, y, z, t) = U (x, z)exp[−2 ln(2)(t/τ )2]exp(−iω0 t)~ey ,

(39)

where τ is the pulse duration and ω0 = 2πc/λ0 is the central frequency. The pulse can be composed in the wave-packet form of a Fourier time expansion (for example, see ref. [58, 59]): ~ H(x, y, z, t) =

Z ∞

~ H(x, z, ω)exp(−iωt)dω.

(40)

−∞

The electric and magnetic fields of the diffracted pulse are found by using the expressions (5-38) for each Fourier’s ω-component of the wave-packet (40). This algorithm is implemented numerically by using the discrete fast Fourier transform (FFT) instead of the integral (40). The spectral interval [ωmin , ωmax ] and the sampling points ωi are optimized by matching the FFT result to the original function (39). The above presented model deals with the incident waves having TM polarization. This polarization is considered for the following reasons. According to the theory of waveguides, the vector wave equations for this polarization can be reduced to one scalar equation describing the magnetic field H of TM modes. The electric component E of these modes is found using the field H and Maxwell’s equations. The reduction simplifies the diffraction problem to one involving a single scalar field in only two dimensions. The TM scalar equation for the component H is decoupled from the similar scalar equation describing the field E of TE (transverse electric) modes. Hence, the formalism works analogously for TE polarization exchanging the E and H fields.

4.

Non-resonant Localization of Continuous Waves and Femtosecond Pulses

In Section 4., we consider the physics of non-resonant transmission and diffraction of continuous waves and femtosecond pulses by a subwavelength nm-size metal slit. The physics is important for understanding the resonant near-field localization of light considered in Section 5.. In the subsection (4.1.)), we investigate the transverse broadening of a continuous wave (a Fourier’s ω-component of a wave-packet) and its connection with the spatial

Resonant Enhancement and Near-Field Localization of FS Pulses...

13

uncertainty ∆x ∼ 1/∆k. The limitations on the transverse dimension and duration of a wave-packet, at a given distance from a subwavelength aperture, imposed by the spatiotemporal uncertainty (Eq. 4) are demonstrated in the subsection (4.2.). The limitations are considered in the context of the spatio-temporal resolution capabilities of subwavelengthbeam optical devices, such as near-field scanning microscopes and spectroscopes. The material used in Section 4. is reprinted from Physics Letters A, Vol. 319, S.V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens 439447, Copyright (2003), with permission from Elsevier, and from Journal of Optics A: Pure and Applied Optics, Vol. 5, 256-262, Copyright (2003), with permission from Institute of Physics Publishing.

4.1.

Non-resonant Near-Field Localization of Continuous Waves

The scattering of light waves by a subwavelength slit is a process that depends on the wavelength λ = 2πc/ω (see, Eqs. 5-38). Owing to the dispersion, the amplitude of a time-harmonic continuous plane wave (a FFT ω-component of a wave-packet) does change under propagation through the slit. At the appropriate conditions, this leads to attenuation or amplification of the wave intensity. The dispersion of a time-harmonic continuous wave is usually described by the normalized transmission coefficient Tcw (λ) for the energy flux ~ = (c/8π)Re(E× ~ H ~ ∗), in CGS units). The coefficient is calculated by integrating the (S normalized energy flux Sz /Szi over the slit value [4, 5]: √ Z a ǫ1 lim [(ExHy∗ + Ex∗Hy )]dx, (41) Tcw = − 4a cos θ −a z→0− where Szi and Sz are the energy fluxes (intensities) of the incident and transmitted waves, respectively. The first objective of our analysis is to check the consistency of the results by comparing the transmission coefficient Tcw = Tcw (λ, a, b) calculated in the studies [4, 5] with those obtained by our computations. We have computed the coefficient for different slit widths 2a and a variety of film thicknesses b. As an example, Fig. 4 shows the transmission coefficient Tcw as a function of screen thickness b for the different slit widths 2 a, at the fixed wavelength λ=500 nm. We notice that the transmission resonances of b = λ/2 periodicity predicted in Refs. [4, 5] are reproduced. The resonance positions and the peak heights ( Tcw ≈λ/2πa) at the resonances are also in agreement with the results [4, 5]. The dispersion function Tcw = Tcw (λ) computed for a continues wave passing through the 50-nm slit (see, Fig. 5 ) also demonstrates the possibility of the resonant enhancement of the wave intensity. Analysis of Figs. 4 and 5 shows that the non-resonant transmission of a continuous wave can be achieved by using a slit in a thin metal film b 2r) of a circular subwavelength aperture in a perfect-conductivity screen the minimum screen thickness b does not depend on the pulse spectra, in contrast to the wavelength-dependent diffraction by a subwavelength slit. In the case of the thin (b < 25 nm) screens, however, the metal films are not completely opaque [71]. This would decrease the spatial and temporal resolutions of a NSOM-pulse system using the thin (b < 25 nm) screens due to passage of the light in the region away from the circular aperture. It should be also noted that in the case of very short ( τ 2r) components exponentially decreases. Due to this effect, the Fourier spectra narrows that leads to the increase of the transmitted pulse duration. The similar effect exists also in the case of the diffraction by a slit in the thin ( b < 25 nm) perfect-conductivity screen. In this case, however, the Fourier spectra narrows due to the amplification of the long-wavelength (λ >> 2r) components of a sub-fs wavepacket (see, curve (A) of Fig. 3). It can be mentioned that for the case λ